Determine the common-mode gain of an op amp that has a differential voltage gain of 150×103 and a CMRR of 90 dB.
A differential amplifier has an open-loop voltage gain of 120 and a common input signal of 3.0 V to both terminals. An output signal of 24 mV results. Calculate the common-mode gain and the CMRR.
The correct answer and explanation is :
Common-Mode Gain and CMRR Calculation for the First Op-Amp
Given:
- Differential voltage gain (( A_d )) = 150,000 (or ( 150 \times 10^3 ))
- CMRR = 90 dB
Step 1: Convert CMRR from dB to a ratio
The CMRR (Common-Mode Rejection Ratio) in dB is related to the ratio of the differential gain (( A_d )) to the common-mode gain (( A_{cm} )):
[
\text{CMRR (dB)} = 20 \log \left( \frac{A_d}{A_{cm}} \right)
]
Rearranging to solve for ( A_{cm} ):
[
A_{cm} = \frac{A_d}{10^{\frac{\text{CMRR (dB)}}{20}}}
]
Substitute the given values:
[
A_{cm} = \frac{150 \times 10^3}{10^{\frac{90}{20}}}
]
[
A_{cm} = \frac{150 \times 10^3}{10^{4.5}}
]
[
A_{cm} = \frac{150 \times 10^3}{31622.8}
]
[
A_{cm} \approx 4.74 \, \text{mV/V}
]
Thus, the common-mode gain for the first op-amp is approximately 4.74 mV/V.
Common-Mode Gain and CMRR Calculation for the Differential Amplifier
Given:
- Open-loop voltage gain (( A_{OL} )) = 120
- Common-mode input voltage = 3.0 V
- Output signal = 24 mV
Step 1: Calculate the common-mode gain
The common-mode gain (( A_{cm} )) can be calculated using the formula:
[
A_{cm} = \frac{V_{out}}{V_{in_cm}}
]
Where ( V_{out} ) is the output signal, and ( V_{in_cm} ) is the common-mode input signal.
Substitute the given values:
[
A_{cm} = \frac{24 \times 10^{-3}}{3.0}
]
[
A_{cm} = 0.008 \, \text{V/V} \, \text{or} \, 8 \, \text{mV/V}
]
Thus, the common-mode gain of the amplifier is 8 mV/V.
Step 2: Calculate the CMRR
The CMRR is calculated as:
[
\text{CMRR} = \frac{A_d}{A_{cm}}
]
Where ( A_d ) is the differential gain, and ( A_{cm} ) is the common-mode gain.
Substitute the values:
[
\text{CMRR} = \frac{120}{8 \times 10^{-3}}
]
[
\text{CMRR} = 15,000
]
To express the CMRR in dB:
[
\text{CMRR (dB)} = 20 \log \left( 15000 \right)
]
[
\text{CMRR (dB)} \approx 82.5 \, \text{dB}
]
Thus, the CMRR of the amplifier is approximately 82.5 dB.
Conclusion
- For the first op-amp, the common-mode gain is 4.74 mV/V.
- For the differential amplifier, the common-mode gain is 8 mV/V, and the CMRR is 82.5 dB.
These calculations show how an op-amp’s ability to reject common-mode signals is quantified, and how high CMRR is crucial for accurate differential amplification.